Observation of entanglement transition of pseudo-random mixed states

Random quantum states serve as a powerful tool in various scientific fields, including quantum supremacy and black hole physics. It has been theoretically predicted that entanglement transitions may happen for different partitions of multipartite random quantum states; however, the experimental observation of these transitions is still absent. Here, we experimentally demonstrate the entanglement transitions witnessed by negativity on a fully connected superconducting processor. We apply parallel entangling operations, that significantly decrease the depth of the pseudo-random circuits, to generate pseudo-random pure states of up to 15 qubits. By quantum state tomography of the reduced density matrix of six qubits, we measure the negativity spectra. Then, by changing the sizes of the environment and subsystems, we observe the entanglement transitions that are directly identified by logarithmic entanglement negativities based on the negativity spectra. In addition, we characterize the randomness of our circuits by measuring the distance between the distribution of output bit-string probabilities and the Porter-Thomas distribution. Our results show that superconducting processors with all-to-all connectivity constitute a promising platform for generating random states and understanding the entanglement structure of multipartite quantum systems.

The manuscript by Liu et al. reports an experiment where a superconducting quantum circuit was used to generate pseudo-random mixed states and to observe phase transitions in their entanglement negativity spectra. For efficient sampling of random quantum states the authors employ a circuit with all-to-all connectivity between the 15 qubits used in experiment, which is mediated by a central bus resonator. The authors calculate negativity spectra from the density matrices of six so-called system qubits, which is measured in experiment by quantum state tomography. The manuscript closely follows a scheme theoretically introduced in Ref. 22 -a generalized theory for entanglement spectra of mixed states in a quantum system that couples to an environment. In the last part of their manuscript, the authors demonstrate the "randomness" of their sampled outputs by observing a Porter-Thomas distribution of the measured output probabilities. As the sampling of random quantum states has been experimentally demonstrated several times before (also with larger quantum circuits and by using different randomization schemes), I view the last part of the paper (Fig. 4) mostly as a necessary prerequisite for the negativity experiment. While the sampling of random quantum states is not novel itself (apart from a possibly new gate sequence for the sampling procedure), the negativity experiment is however novel to my knowledge. The manuscript is written concisely, with the relevant theoretical background information introduced. The experimental results show good agreement with theoretical expectations and numerical simulations. The topic of the manuscript is timely and interesting to a general readership. In my opinion, the manuscript can therefore be a good fit for Nature Communications. Before I can give a final recommendation I would like to ask the authors to clarify several points as detailed in the following. The superconducting device used in this manuscript looks very similar to the one used in Ref 1. The authors mention in the text and in their abstract that their chip contains 20 transmon qubits. I recommend that the authors clearly state in their text that they use only 15 for their experiment, which is implied in the second sentence of the Results section. A comment on the frequency location of the five remaining qubits during the experiment would help to clarify (already in the first paragraph of Results). For the abstract, I suggest to only mention that 15 qubits are used in the experiment, since the third sentence may imply that all 20 qubits have been utilized. According to Fig. 1c, the coupling strength between qubits is on the order of 0.5MHz to 2MHz. Can the authors provide some intuition as to how effective the 40ns free evolution per layer is to spread entanglement within the qubit lattice? The authors mention that the rotation angles for the single-qubit rotations are "uniformly sampled" from certain intervals. I would appreciate some more details on this procedure. Are the rotation angles themselves random? If not, can the authors discuss how this may impact the randomness of the measurement outcomes? In addition, I encourage the authors to discuss infidelities in the single-qubit rotation gates and variations in the qubit-qubit coupling strengths. Is the gate fidelity known or calibrated? I can imagine that certain infidelities may not harm the randomization process, but certain others which may be the result of unwanted correlations may influence the experimental outcome. Are interactions between qubits turned off during the single-qubit gates? Similarly, I wonder how differences or changes in qubit-qubit coupling strengths impact this experiment. Have the authors selected always the same six qubits as system qubits? Since the coupling map in Fig. 1c shows non-identical couplings between different pairs of qubits it might be a good sanity check to choose different qubits as the system qubits in successive experimental runs in order to support that variations in coupling strengths or gate infidelities do not impact the measurement outcome. In Fig. 2a the authors show the phase diagram for infinitely large N, and it is my understanding that it was derived in Ref. 22 for the large-system limit. Can the authors comment on the validity of this phase diagram for finite system (and environment) sizes, as used in their experiment? The authors mention that they used five layers for their experiments in Fig. 2 and Fig. 3. However, in Fig. 4 they show that the experimentally extracted probability distributions are closest to the expected Porter-Thomas distribution for three layers. Can the authors please elaborate on this discrepancy? This suggests that the underlying probability distributions used for the negativity results do not follow a Porter-Thomas distribution. Logically, I would rather check that the probability distribution follows Porter-Thomas first (and find the optimal number of layers based on that) and then use this optimal layer number for the negativity experiments (basically show Fig. 4 first, followed by the negativity results). Can the authors comment why they did not adopt this strategy? The authors do not mention the anharmonicity of their qubits nor the effect of the second excited levels of their transmon qubits. Did the authors check the higher-level excitations or can they motivate that there is no adverse effect for the probability distribution measurements, density matrix extraction, or their experimental scheme in general? Since the experimental realization of quantum state tomography in a six-qubit system is rather expensive in terms of number of measurements I would appreciate if the authors could provide some details on their experimental methods. I was not able to find the Supplementary Notes, where I assume the authors have made more explanations on the experimental details, but in any case I would appreciate it if the authors could add a few additional explanatory sentences on their experimental procedure in the main text or in the Methods section. This also applies to their correction scheme of data points, which deserves at least a short explanation in the main text. In Fig. 3a, why is the negativity for N_B=7 so small, when according to the text we would expect nonzero negativity in the NPT region? Might a logarithmic scaling of the vertical axis in Fig. 3a help to better visualize these (small) values? The authors explain how they divide their circuit into three parts A1, A2, and B in the last paragraph of page 3, but already use this notation in the context of negativity. Fig 1a is missing a scale bar. In Fig. 1c the authors should invert their color bar, such that highest coupling strengths are not white.

Reviewer #2 (Remarks to the Author):
The work titled Observation of entanglement transition of pseudo-random mixed states by Tong Liu and coworkers have studied the Haar measure random states generated using superconducting processor. They have demonstrated the entanglement transition as predicted by works published in Refs. [19,20,21,22]. This is a very important experimental demonstration of its kind. The experimental results agrees very well the theoretical once considering the error introducing factors like decoherence. This system will also act as a testbed for testing other related results. The paper is well written with all key results explained very well. The figures are comprehensible. Key papers are included in the references. Despite these positives, I think authors need to implement following suggestions: 1. The authors should specify the total number of realizations i.e. total number of states generated (experiments and simulations) to get the plots of figures 2, 3 and 4. Also the time taken to generated one full state and its QST.
2. Some important references on QST can be added since it is one of the main tool in this work.

Authors can add the following relevant reference in the introduction: Experimental Realization of a Measurement-Induced Entanglement Phase Transition on a Superconducting Quantum
Processor by Jin Ming Koh, Shi-Ning Sun, Mario Motta, and Austin J. Minnich (https://arxiv.org/abs/2203.04338).
Finally, I feel that the authors have experimentally demonstrated an important theoretical result on the entanglement transition. This work deserves the publication in Nature Communications and I am happy to recommend this work for publication.

Reviewer #3 (Remarks to the Author):
This work is a very nice experimental study of transitions in the structure of entanglement in quantum states generated by pseudorandom circuits. The novel experimental aspect of this study is the use of nonlocal connectivity, which allows for rapid scrambling. The reduced density matrices are characterized by standard tomographic techniques (and hence so is the negativity). The bit-string probabilities are also computed and found to agree with expectations for random circuits.
This work performs a nontrivial experimental test of an interesting theory prediction and as such I recommend its publication. The paper is clearly written and I have no substantive criticisms.